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<div class="header">
  <div class="summary">
<a href="#var-members">Variables</a>  </div>
  <div class="headertitle">
<div class="title">Comonad<div class="ingroups"><a class="el" href="group__group-concepts.html">Concepts</a></div></div>  </div>
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<div class="contents">
<a name="details" id="details"></a><h2 class="groupheader">Description</h2>
<p>The <code>Comonad</code> concept represents context-sensitive computations and data. </p>
<p>Formally, the Comonad concept is dual to the Monad concept. But unless you're a mathematician, you don't care about that and it's fine. So intuitively, a Comonad represents context sensitive values and computations. First, Comonads make it possible to extract context-sensitive values from their context with <code>extract</code>. In contrast, Monads make it possible to wrap raw values into a given context with <code>lift</code> (from Applicative).</p>
<p>Secondly, Comonads make it possible to apply context-sensitive values to functions accepting those, and to return the result as a context-sensitive value using <code>extend</code>. In contrast, Monads make it possible to apply a monadic value to a function accepting a normal value and returning a monadic value, and to return the result as a monadic value (with <code>chain</code>).</p>
<p>Finally, Comonads make it possible to wrap a context-sensitive value into an extra layer of context using <code>duplicate</code>, while Monads make it possible to take a value with an extra layer of context and to strip it with <code>flatten</code>.</p>
<p>Whereas <code>lift</code>, <code>chain</code> and <code>flatten</code> from Applicative and Monad have signatures </p><p class="formulaDsp">
\begin{align*} \mathtt{lift}_M &amp;: T \to M(T) \\ \mathtt{chain} &amp;: M(T) \times (T \to M(U)) \to M(U) \\ \mathtt{flatten} &amp;: M(M(T)) \to M(T) \end{align*}
</p>
<p><code>extract</code>, <code>extend</code> and <code>duplicate</code> from Comonad have signatures </p><p class="formulaDsp">
\begin{align*} \mathtt{extract} &amp;: W(T) \to T \\ \mathtt{extend} &amp;: W(T) \times (W(T) \to U) \to W(U) \\ \mathtt{duplicate} &amp;: W(T) \to W(W(T)) \end{align*}
</p>
<p>Notice how the "arrows" are reversed. This symmetry is essentially what we mean by Comonad being the <em>dual</em> of Monad.</p>
<dl class="section note"><dt>Note</dt><dd>The <a href="https://wiki.haskell.org/Typeclassopedia#Comonad">Typeclassopedia</a> is a nice Haskell-oriented resource for further reading about Comonads.</dd></dl>
<h2>Minimal complete definition </h2>
<p><code>extract</code> and (<code>extend</code> or <code>duplicate</code>) satisfying the laws below. A <code>Comonad</code> must also be a <code>Functor</code>.</p>
<h2>Laws </h2>
<p>For all Comonads <code>w</code>, the following laws must be satisfied: </p><div class="fragment"><div class="line"><a class="code" href="group__group-Comonad.html#ga307479a91a21b7ab06a2bc746b003dcc">extract</a>(<a class="code" href="group__group-Comonad.html#ga181751278bd19a4bfc3c08bd7ddef399">duplicate</a>(w)) == w</div><div class="line"><a class="code" href="group__group-Functor.html#ga5a8975f6e55375d5b6038a9c36ee63e7">transform</a>(<a class="code" href="group__group-Comonad.html#ga181751278bd19a4bfc3c08bd7ddef399">duplicate</a>(w), <a class="code" href="group__group-Comonad.html#ga307479a91a21b7ab06a2bc746b003dcc">extract</a>) == w</div><div class="line"><a class="code" href="group__group-Comonad.html#ga181751278bd19a4bfc3c08bd7ddef399">duplicate</a>(<a class="code" href="group__group-Comonad.html#ga181751278bd19a4bfc3c08bd7ddef399">duplicate</a>(w)) == <a class="code" href="group__group-Functor.html#ga5a8975f6e55375d5b6038a9c36ee63e7">transform</a>(<a class="code" href="group__group-Comonad.html#ga181751278bd19a4bfc3c08bd7ddef399">duplicate</a>(w), <a class="code" href="group__group-Comonad.html#ga181751278bd19a4bfc3c08bd7ddef399">duplicate</a>)</div></div><!-- fragment --><dl class="section note"><dt>Note</dt><dd>There are several equivalent ways of defining Comonads, and this one is just one that was picked arbitrarily for simplicity.</dd></dl>
<h2>Refined concept </h2>
<ol type="1">
<li>Functor<br />
Every Comonad is also required to be a Functor. At first, one might think that it should instead be some imaginary concept CoFunctor. However, it turns out that a CoFunctor is the same as a <code>Functor</code>, hence the requirement that a <code>Comonad</code> also is a <code>Functor</code>.</li>
</ol>
<h2>Concrete models </h2>
<p><code><a class="el" href="structboost_1_1hana_1_1lazy.html" title="hana::lazy implements superficial laziness via a monadic interface. ">hana::lazy</a></code></p>
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Variables</h2></td></tr>
<tr class="memitem:ga181751278bd19a4bfc3c08bd7ddef399"><td class="memItemLeft" align="right" valign="top">constexpr auto&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__group-Comonad.html#ga181751278bd19a4bfc3c08bd7ddef399">boost::hana::duplicate</a></td></tr>
<tr class="memdesc:ga181751278bd19a4bfc3c08bd7ddef399"><td class="mdescLeft">&#160;</td><td class="mdescRight">Add an extra layer of comonadic context to a comonadic value.Given a value already in a comonadic context, <code>duplicate</code> wraps this value with an additional layer of comonadic context. This can be seen as the dual operation to <code>flatten</code> from the Monad concept.  <a href="group__group-Comonad.html#ga181751278bd19a4bfc3c08bd7ddef399">More...</a><br /></td></tr>
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<tr class="memitem:gaf44692351fd9fe4e76815dfef6ff4478"><td class="memItemLeft" align="right" valign="top">constexpr auto&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="group__group-Comonad.html#gaf44692351fd9fe4e76815dfef6ff4478">boost::hana::extend</a></td></tr>
<tr class="memdesc:gaf44692351fd9fe4e76815dfef6ff4478"><td class="mdescLeft">&#160;</td><td class="mdescRight">Comonadic application of a function to a comonadic value.Given a comonadic value and a function accepting a comonadic input, <code>extend</code> returns the result of applying the function to that input inside the comonadic context.  <a href="group__group-Comonad.html#gaf44692351fd9fe4e76815dfef6ff4478">More...</a><br /></td></tr>
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<tr class="memdesc:ga307479a91a21b7ab06a2bc746b003dcc"><td class="mdescLeft">&#160;</td><td class="mdescRight">Extract a value in a given comonadic context.Given a value inside a comonadic context, extract it from that context, performing whatever effects are mandated by that context. This can be seen as the dual operation to the <code>lift</code> method of the Applicative concept.  <a href="group__group-Comonad.html#ga307479a91a21b7ab06a2bc746b003dcc">More...</a><br /></td></tr>
<tr class="separator:ga307479a91a21b7ab06a2bc746b003dcc"><td class="memSeparator" colspan="2">&#160;</td></tr>
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<h2 class="groupheader">Variable Documentation</h2>
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<p><code>#include &lt;<a class="el" href="fwd_2duplicate_8hpp.html">boost/hana/fwd/duplicate.hpp</a>&gt;</code></p>
<b>Initial value:</b><div class="fragment"><div class="line">= [](<span class="keyword">auto</span>&amp;&amp; w) -&gt; decltype(<span class="keyword">auto</span>) {</div><div class="line">        <span class="keywordflow">return</span> tag-dispatched;</div><div class="line">    }</div></div><!-- fragment -->
<p>Add an extra layer of comonadic context to a comonadic value.Given a value already in a comonadic context, <code>duplicate</code> wraps this value with an additional layer of comonadic context. This can be seen as the dual operation to <code>flatten</code> from the Monad concept. </p>
<h2>Signature </h2>
<p>Given a Comonad <code>W</code>, the signature is \( \mathtt{duplicate} : W(T) \to W(W(T)) \)</p>
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">w</td><td>The value to wrap in an additional level of comonadic context.</td></tr>
  </table>
  </dd>
</dl>
<h2>Example </h2>
<div class="fragment"><div class="line"><span class="comment">// Copyright Louis Dionne 2013-2016</span></div><div class="line"><span class="comment">// Distributed under the Boost Software License, Version 1.0.</span></div><div class="line"><span class="comment">// (See accompanying file LICENSE.md or copy at http://boost.org/LICENSE_1_0.txt)</span></div><div class="line"></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="duplicate_8hpp.html">boost/hana/duplicate.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="extract_8hpp.html">boost/hana/extract.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="lazy_8hpp.html">boost/hana/lazy.hpp</a>&gt;</span></div><div class="line"><span class="keyword">namespace </span>hana = <a class="code" href="namespaceboost_1_1hana.html">boost::hana</a>;</div><div class="line"></div><div class="line"></div><div class="line"><span class="keywordtype">int</span> main() {</div><div class="line">    constexpr <span class="keyword">auto</span> very_lazy = <a class="code" href="group__group-Comonad.html#ga181751278bd19a4bfc3c08bd7ddef399">hana::duplicate</a>(hana::make_lazy(3));</div><div class="line">    static_assert(<a class="code" href="group__group-Comonad.html#ga307479a91a21b7ab06a2bc746b003dcc">hana::extract</a>(<a class="code" href="group__group-Comonad.html#ga307479a91a21b7ab06a2bc746b003dcc">hana::extract</a>(very_lazy)) == 3, <span class="stringliteral">&quot;&quot;</span>);</div><div class="line">}</div></div><!-- fragment -->
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<p><code>#include &lt;<a class="el" href="fwd_2extend_8hpp.html">boost/hana/fwd/extend.hpp</a>&gt;</code></p>
<b>Initial value:</b><div class="fragment"><div class="line">= [](<span class="keyword">auto</span>&amp;&amp; w, <span class="keyword">auto</span>&amp;&amp; f) -&gt; decltype(<span class="keyword">auto</span>) {</div><div class="line">        <span class="keywordflow">return</span> tag-dispatched;</div><div class="line">    }</div></div><!-- fragment -->
<p>Comonadic application of a function to a comonadic value.Given a comonadic value and a function accepting a comonadic input, <code>extend</code> returns the result of applying the function to that input inside the comonadic context. </p>
<h2>Signature </h2>
<p>Given a Comonad <code>W</code> and a function of type \( W(T) \to U \), the signature is \( \mathtt{extend} : W(T) \times (W(T) \to U) \to W(U) \)</p>
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">w</td><td>A comonadic value to call the function with.</td></tr>
    <tr><td class="paramname">f</td><td>A function of signature \( W(T) \to U \) to be applied to its comonadic argument inside the comonadic context.</td></tr>
  </table>
  </dd>
</dl>
<h2>Example </h2>
<div class="fragment"><div class="line"><span class="comment">// Copyright Louis Dionne 2013-2016</span></div><div class="line"><span class="comment">// Distributed under the Boost Software License, Version 1.0.</span></div><div class="line"><span class="comment">// (See accompanying file LICENSE.md or copy at http://boost.org/LICENSE_1_0.txt)</span></div><div class="line"></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="assert_8hpp.html">boost/hana/assert.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="eval_8hpp.html">boost/hana/eval.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="extend_8hpp.html">boost/hana/extend.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="extract_8hpp.html">boost/hana/extract.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="lazy_8hpp.html">boost/hana/lazy.hpp</a>&gt;</span></div><div class="line"></div><div class="line"><span class="preprocessor">#include &lt;functional&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;istream&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;sstream&gt;</span></div><div class="line"><span class="keyword">namespace </span>hana = <a class="code" href="namespaceboost_1_1hana.html">boost::hana</a>;</div><div class="line"></div><div class="line"></div><div class="line"><span class="keyword">template</span> &lt;<span class="keyword">typename</span> T&gt;</div><div class="line">T read_one(std::istream&amp; s) {</div><div class="line">    T <a class="code" href="group__group-Constant.html#ga1687520692a6b0c49e3a69de2980f388">value</a>;</div><div class="line">    s &gt;&gt; <a class="code" href="group__group-Constant.html#ga1687520692a6b0c49e3a69de2980f388">value</a>;</div><div class="line">    <span class="keywordflow">return</span> <a class="code" href="group__group-Constant.html#ga1687520692a6b0c49e3a69de2980f388">value</a>;</div><div class="line">}</div><div class="line"></div><div class="line"><span class="keywordtype">int</span> main() {</div><div class="line">    std::stringstream s;</div><div class="line">    s &lt;&lt; <span class="stringliteral">&quot;1 2 3&quot;</span>;</div><div class="line"></div><div class="line">    <span class="keyword">auto</span> from_stream = <a class="code" href="group__group-Comonad.html#gaf44692351fd9fe4e76815dfef6ff4478">hana::extend</a>(hana::make_lazy(read_one&lt;int&gt;)(std::ref(s)), [](<span class="keyword">auto</span> i) {</div><div class="line">        <span class="keywordflow">return</span> hana::eval(i) + 1;</div><div class="line">    });</div><div class="line"></div><div class="line">    <a class="code" href="group__group-assertions.html#ga29b2b21ffa5513e5b706c50ffee980af">BOOST_HANA_RUNTIME_CHECK</a>(<a class="code" href="group__group-Comonad.html#ga307479a91a21b7ab06a2bc746b003dcc">hana::extract</a>(from_stream) == 2);</div><div class="line">    <a class="code" href="group__group-assertions.html#ga29b2b21ffa5513e5b706c50ffee980af">BOOST_HANA_RUNTIME_CHECK</a>(<a class="code" href="group__group-Comonad.html#ga307479a91a21b7ab06a2bc746b003dcc">hana::extract</a>(from_stream) == 3);</div><div class="line">    <a class="code" href="group__group-assertions.html#ga29b2b21ffa5513e5b706c50ffee980af">BOOST_HANA_RUNTIME_CHECK</a>(<a class="code" href="group__group-Comonad.html#ga307479a91a21b7ab06a2bc746b003dcc">hana::extract</a>(from_stream) == 4);</div><div class="line">}</div></div><!-- fragment -->
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<p><code>#include &lt;<a class="el" href="fwd_2extract_8hpp.html">boost/hana/fwd/extract.hpp</a>&gt;</code></p>
<b>Initial value:</b><div class="fragment"><div class="line">= [](<span class="keyword">auto</span>&amp;&amp; w) -&gt; decltype(<span class="keyword">auto</span>) {</div><div class="line">        <span class="keywordflow">return</span> tag-dispatched;</div><div class="line">    }</div></div><!-- fragment -->
<p>Extract a value in a given comonadic context.Given a value inside a comonadic context, extract it from that context, performing whatever effects are mandated by that context. This can be seen as the dual operation to the <code>lift</code> method of the Applicative concept. </p>
<h2>Signature </h2>
<p>Given a Comonad <code>W</code>, the signature is \( \mathtt{extract} : W(T) \to T \)</p>
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">w</td><td>The value to be extracted inside a comonadic context.</td></tr>
  </table>
  </dd>
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<h2>Example </h2>
<div class="fragment"><div class="line"><span class="comment">// Copyright Louis Dionne 2013-2016</span></div><div class="line"><span class="comment">// Distributed under the Boost Software License, Version 1.0.</span></div><div class="line"><span class="comment">// (See accompanying file LICENSE.md or copy at http://boost.org/LICENSE_1_0.txt)</span></div><div class="line"></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="extract_8hpp.html">boost/hana/extract.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="placeholder_8hpp.html">boost/hana/functional/placeholder.hpp</a>&gt;</span></div><div class="line"><span class="preprocessor">#include &lt;<a class="code" href="lazy_8hpp.html">boost/hana/lazy.hpp</a>&gt;</span></div><div class="line"><span class="keyword">namespace </span>hana = <a class="code" href="namespaceboost_1_1hana.html">boost::hana</a>;</div><div class="line"></div><div class="line"></div><div class="line">static_assert(<a class="code" href="group__group-Comonad.html#ga307479a91a21b7ab06a2bc746b003dcc">hana::extract</a>(hana::make_lazy(1)) == 1, <span class="stringliteral">&quot;&quot;</span>);</div><div class="line">static_assert(<a class="code" href="group__group-Comonad.html#ga307479a91a21b7ab06a2bc746b003dcc">hana::extract</a>(hana::make_lazy(<a class="code" href="group__group-functional.html#gaefe9fd152cba94be71c2b5b9de689d23">hana::_</a> + 1)(3)) == 4, <span class="stringliteral">&quot;&quot;</span>);</div><div class="line"></div><div class="line"><span class="keywordtype">int</span> main() { }</div></div><!-- fragment -->
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